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Mirrors > Home > MPE Home > Th. List > php2 | Structured version Visualization version GIF version |
Description: Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) |
Ref | Expression |
---|---|
php2 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2903 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ω ↔ 𝐴 ∈ ω)) | |
2 | psseq2 4068 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ⊊ 𝑥 ↔ 𝐵 ⊊ 𝐴)) | |
3 | 1, 2 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ω ∧ 𝐵 ⊊ 𝑥) ↔ (𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴))) |
4 | breq2 5073 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ≺ 𝑥 ↔ 𝐵 ≺ 𝐴)) | |
5 | 3, 4 | imbi12d 347 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ ω ∧ 𝐵 ⊊ 𝑥) → 𝐵 ≺ 𝑥) ↔ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴))) |
6 | vex 3500 | . . . . . 6 ⊢ 𝑥 ∈ V | |
7 | pssss 4075 | . . . . . 6 ⊢ (𝐵 ⊊ 𝑥 → 𝐵 ⊆ 𝑥) | |
8 | ssdomg 8558 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝐵 ⊆ 𝑥 → 𝐵 ≼ 𝑥)) | |
9 | 6, 7, 8 | mpsyl 68 | . . . . 5 ⊢ (𝐵 ⊊ 𝑥 → 𝐵 ≼ 𝑥) |
10 | 9 | adantl 484 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ⊊ 𝑥) → 𝐵 ≼ 𝑥) |
11 | php 8704 | . . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ⊊ 𝑥) → ¬ 𝑥 ≈ 𝐵) | |
12 | ensym 8561 | . . . . 5 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵) | |
13 | 11, 12 | nsyl 142 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ⊊ 𝑥) → ¬ 𝐵 ≈ 𝑥) |
14 | brsdom 8535 | . . . 4 ⊢ (𝐵 ≺ 𝑥 ↔ (𝐵 ≼ 𝑥 ∧ ¬ 𝐵 ≈ 𝑥)) | |
15 | 10, 13, 14 | sylanbrc 585 | . . 3 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ⊊ 𝑥) → 𝐵 ≺ 𝑥) |
16 | 5, 15 | vtoclg 3570 | . 2 ⊢ (𝐴 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)) |
17 | 16 | anabsi5 667 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 ⊊ wpss 3940 class class class wbr 5069 ωcom 7583 ≈ cen 8509 ≼ cdom 8510 ≺ csdm 8511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 |
This theorem is referenced by: php4 8707 nndomo 8715 nndomog 39903 |
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